Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group
Abstract
We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haardistributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson Zuber type.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2006
 DOI:
 10.1007/s0022000615543
 arXiv:
 arXiv:mathph/0402073
 Bibcode:
 2006CMaPh.264..773C
 Keywords:

 Mathematical Physics;
 Mathematics  Probability;
 28C10;
 22E30;
 46L54
 EPrint:
 Commun. Math. Phys. 264 (2006), no. 3, 773795